From aa8c1818013a577f147e5aa23e5c86a0e9e9da97 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Fri, 23 Sep 2022 22:58:18 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../derivatives-and-differential.pdf | Bin 143261 -> 143253 bytes .../differential-equation.pdf | Bin 125552 -> 125706 bytes .../differential-equation.tex | 10 ++- .../mathematical-statistics.pdf | Bin 104211 -> 114167 bytes .../mathematical-statistics.tex | 52 ++++++++++- .../mathematical-statistics.pdf | Bin 158496 -> 160775 bytes .../mathematical-statistics.tex | 83 +++++++++++------- 7 files changed, 106 insertions(+), 39 deletions(-) diff --git a/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf b/advanced-math/knowledge/2-derivatives-and-differential/derivatives-and-differential.pdf index 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25ae257..487a7bf 100644 --- a/probability-theory-and-mathematical-statistics/exercise/5-mathematical-statistics/mathematical-statistics.tex +++ b/probability-theory-and-mathematical-statistics/exercise/5-mathematical-statistics/mathematical-statistics.tex @@ -122,6 +122,12 @@ $\therefore P\{X^2>C^2\}=0.8$。 基本方法就是$EX=\dfrac{1}{n}\sum\limits_{i=1}^nX_i$。 +如果只有一个参数就使用一阶矩,如果有两个参数就使用二阶矩,一般不会超过两个未知数。 + +即$EX=\dfrac{1}{n}\sum\limits_{i=1}^nX_i=\overline{X}=\hat{\mu}$,$EX^2=\dfrac{1}{n}\sum\limits_{i=1}^nX_i^2=\hat{\sigma}^2+\hat{\mu}^2$。 + +$\hat{\sigma}^2=EX^2-(EX)^2=\dfrac{1}{n}\sum\limits_{i=1}^nX_i^2-\overline{X}^2=\dfrac{1}{n}\sum\limits_{i=1}^n(X_i-\overline{X})^2$。 + \subsubsection{一阶矩} \subsubsection{二阶矩} @@ -138,7 +144,39 @@ $\therefore P\{X^2>C^2\}=0.8$。 \subsection{最大似然估计} -步骤:写出概率函数或密度函数;写出似然函数(代入观测值$x_i$并连乘);两边取对数;求导数并令为0。 +步骤:写出概率函数或密度函数;写出似然函数(代入观测值$x_i$并连乘);两边取对数;求导数并令为0求出表达式。 + +\subsubsection{离散型} + +\textbf{例题:}设总体$X$的概率分布为:\medskip + +\begin{tabular}{c|cccc} + \hline + $X$ & 0 & 1 & 2 & 3 \\ \hline + $P$ & $\theta^2$ & $2\theta(1-\theta)$ & $\theta^2$ & $1-2\theta$ \\ \hline +\end{tabular} \medskip + +其中$\theta\int\left(0,\dfrac{1}{2}\right)$为未知参数,从总体$X$中抽取容量为8的一组样本,其样本值为3,1,3,0,3,1,2,3。求$\theta$的最大似然估计值。 + +解: + +根据样本值,可以得出:\medskip + +\begin{tabular}{c|cccc} + \hline + $X$ & 0 & 1 & 2 & 3 \\ \hline + 次数 & 1 & 2 & 1 & 4 \\ \hline +\end{tabular} \medskip + +将所有的概率相乘:$L(\theta)l=(1-2\theta)^4[2\theta(1-\theta)]^2\cdot\theta^2\cdot\theta^2=4\theta^6(1-\theta)^2(1-2\theta)^4$。 + +对其求对数:$\ln L(\theta)=\ln4+6\ln\theta+2\ln(1-\theta)+4\ln(1-2\theta)$。 + +对其求导:$\dfrac{\textrm{d}\ln L(\theta)}{\textrm{d}\theta}=\dfrac{6}{\theta}-\dfrac{2}{1-\theta}-\dfrac{8}{1-2\theta}=0$。解得$\theta=\dfrac{7\pm\sqrt{13}}{12}$。 + +$0<\theta<\dfrac{1}{2}$,舍去正值,得到$\hat{\theta}=\dfrac{7-\sqrt{13}}{12}$。 + +\subsubsection{连续型} \textbf{例题:}设随机变量$X$在区间$[0,\theta]$上服从均匀分布,$X_1,X_2,\cdots,X_n$是来自$X$的简单随机样本,求$\theta$的最大似然估计量$\hat{\theta}$ @@ -166,6 +204,18 @@ $\ln 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a/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex b/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex index 28152ac..f4c1fc1 100644 --- a/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex +++ b/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex @@ -302,24 +302,6 @@ $\therefore P\{F>F_\alpha(n_2,n_1)\}=\alpha$,$P\{F\leqslant F_\alpha(n_2,n_1)\ 即将概率密度或概率分布连乘,然后取对数,再求导令其为0解出$\hat{\theta}$。 -\textbf{例题:}设总体$X$的概率分布为:\medskip - -\begin{tabular}{c|cccc} - \hline - $X$ & 0 & 1 & 2 & 3 \\ \hline - $P$ & $\theta^2$ & $2\theta(1-\theta)$ & $\theta^2$ & $1-2\theta$ \\ \hline -\end{tabular} \medskip - -其中$\theta\int\left(0,\dfrac{1}{2}\right)$为未知参数,从总体$X$中抽取容量为8的一组样本,其样本值为3,1,3,0,3,1,2,3。求$\theta$的矩估计值和最大似然估计值。 - -解:首先将所有的概率相乘:$L(\theta)l=(1-2\theta)^4[2\theta(1-\theta)]^2\cdot\theta^2\cdot\theta^2=4\theta^6(1-\theta)^2(1-2\theta)^4$。 - -对其求对数:$\ln L(\theta)=\ln4+6\ln\theta+2\ln(1-\theta)+4\ln(1-2\theta)$。 - -对其求导:$\dfrac{\textrm{d}\ln L(\theta)}{\textrm{d}\theta}=\dfrac{6}{\theta}-\dfrac{2}{1-\theta}-\dfrac{8}{1-2\theta}=0$。解得$\theta=\dfrac{7\pm\sqrt{13}}{12}$。 - -$0<\theta<\dfrac{1}{2}$,舍去正值,得到$\hat{\theta}=\dfrac{7-\sqrt{13}}{12}$。 - \textbf{例题:}来自总体的$X$的概率密度为$f(x)=\left\{\begin{array}{ll} (1+\theta)x^\theta, & 0\theta_0$ \\ \cline{2-4} & 左边 & $\theta\geqslant\theta_0$ & $\theta<\theta_0$ \\ \hline -\end{tabular} +\end{tabular} \medskip + +假设检验步骤: + +\begin{enumerate} + \item 根据问题要求提出原假设$H_0$和备择假设$H_1$。 + \item 根据假设和条件确定检验统计量,在$H_0$成立的条件下确定其分布。 + \item 给定显著性水平$\alpha$,在$H_0$成立的条件下根据$P\{H_0\text{为真,拒绝}H_0\}\leqslant\alpha$确定拒绝域和临界点。 + \item 由样本值计算出检验统计量值,若该值落入拒绝域,则拒绝$H_0$,否则接受$H_0$。 +\end{enumerate} \subsubsection{正态总体下的六大检验与拒绝域}